I hope people will indulge me a newbie question on notation.
For reference, the textbook in question is Contemporary Abstract Algebra by Joseph Gallian (8th edition).
I was reading the chapter on extension fields, which defines the Fundamental Theorem of Field Theory as follows: "Let F be a field and let f(x) be a nonconstant polynomial in F[x]. Then there is an extension field E of F in which f(x) has a zero." It then gives the following example:
Let $f(x)=x^2 + 1 \in Q[x]$. Then, viewing $f(x)$ as an element of $E[x] = (Q[x]/\langle x^2 + 1\rangle)[x]$, we have $$ f(x + \langle x^2 + 1 \rangle) = (x + \langle x^2 + 1 \rangle)^2 + 1 = x^2 + \langle x^2 + 1 \rangle + 1 = x^2 + 1 + \langle x^2 + 1 \rangle = 0 + \langle x^2 + 1 \rangle $$
I have to admit I'm slightly baffled by this example and his notation. So I understand that $Q[x]$ is a polynomial ring with rational coefficients, that $G/H$ is a factor/quotient ring (or a factor group in group theory), and that $\langle a \rangle$ is an ideal generated by $a$ (or a cyclic group generated by $a$ in group theory), but I'm having trouble understanding what this actually means when put together. Can someone help me understand this example? I'm having particular difficulty understanding what $E[x] = (Q[x]/\langle x^2 + 1\rangle)[x]$ means. What does this actually consist of (i.e. what are its elements)?